Serendipity with Center of Mass Demo

Recently I set up a demonstration that I hadn’t done for a while. You balance a dynamic track on a thin board. You then place two carts with different masses on it so the center of mass of the two-cart system is directly over the thin board that acts like a fulcrum. The track stays balanced. I used to tie the carts close together with a thread. Their magnets would repel and send the carts flying apart when I burned the thread with a match. This was a little tricky to set up and difficult to repeat quickly if something went wrong. I decided to try the spring-loaded plungers built into the carts. I avoided this before because it requires touching the carts to release the plunger.

It turns out this wasn’t a problem, the carts still separated cleanly. This made doing the demo a lot easier. I was in my room on the weekend trying this out and decided to make a video of the demo in slow-motion. This would be good for discussion and as a back-up if things went wrong. I marked the initial position of the center of mass of the carts with a white sticker. The blue cart is 0.75 kg and the red cart is 0.25 kg. Here is the result in 300 frames per second:

After viewing the video I was thinking about how it showed that the internal force of the plunger did not affect the velocity of the center of mass. It was zero before the separation and zero up to the point an external force acts on the red cart hitting the end of the track.

I thought if I could get the carts to move back toward each other with the same speed, the track would stay balanced too. This would be difficult to arrange in reality, but with video editing, it was easy. I imported the video into iMovie and used the rewind feature. Now it shows the two carts heading toward each other, compressing the spring, then flying back out. The track stays balanced the whole time, even with the excited physics teacher flapping his arms around and poking the blue cart with a pencil during the collisions, see for yourself:

This is what every one-dimensional elastic collision looks like if the velocity of the center of mass is zero from your frame of reference. There is a well-known technique exploiting this fact that makes solving for the 2 final velocities of an elastic collision much easier algebraically than using conservation of momentum and energy. I have shown this technique to my students before but felt they really didn’t understand why it worked.

This video seemed to be a good tool to try and improve their understanding. They see that if you are in the center of mass' frame, in an elastic collision the carts change direction after the collision, retaining their original speed. All you need to do is first subtract the velocity of the center of mass from their initial velocities to put yourself in the frame of reference where vcm is zero.

After switching the direction of the speeds, you need to go back to the rest frame by adding it back to the final velocities. The video gives the algorithm for doing this more meaning for the students:

1. Find vcm, it equals the total momentum divided by the total mass.
2. Subtract vcm from the initial velocities.
3. Change the sign of the result. You now have the final velocities in the center of mass' frame of reference.
4. Add vcm to the final velocities, you now have the final final velocities in the rest frame of reference.

I suggest having students use video analysis to implement this technique. I still want my students to be able to write out the equations for conservation of momentum and energy. They can do it after using the center of mass technique to check their results.